Let $X$ be a smooth, projective, geometrically connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every action of a finite $p$-group on $X$ factors through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer properties of representations of finite $p$-groups to their actions on $X$. We deduce a fixed-point theorem which, unlike previously known results of this kind, is sensitive to the arithmetic of the base field. We also obtain a bound on the orders of cyclic $p$-subgroups of the Cremona groups: for instance $\operatorname{Cr}_n(\mathbb{Q})$ contains no element of order $p^2$ when $p \ge n+2$. The method is based on the following observation, of independent interest. For an affine algebraic group $G$ over a field of characteristic zero, $\operatorname{ed}_p(G) + \dim G$ is the least dimension of a smooth projective variety $Y$ with a generically free $G$-action such that the degree map $\operatorname{CH}_G(Y) \to \mathbb{F}_p$ is nonzero. A key input for our result is Karpenko and Merkurjev's computation of the essential $p$-dimension of $p$-groups.
Let $K$ be an imaginary quadratic number field, let $p$ be an odd prime, and let $G=G_{\varnothing}(K)(p)$ be the Galois group of the maximal everywhere unramified pro-$p$ extension of $K$. To each mod-$p$ character $x$ of $G$ we associate a linear map $D_x$ from $\mathrm{Cl}(K)[p]$ to $\mathrm{Cl}(K)/p$; a formula of...
Let $X$ be a curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$, and let $G=\mathrm{Aut}(X)$. A theorem of Giulietti and Korchmaros shows that if $G$ fixes no point and a $p$-subgroup $S\le G_P$ satisfies $|S|>\frac{p}{p-1}g$, then $X$ is one of the classical hyperelliptic, Hermitian, o...
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $C_p$ act linearly on a finite-dimensional $k$-vector space $V$, with indecomposable Jordan summands $V_{d_i}$ of dimension $d_i$. We study projective (crepant) resolutions of $V/C_p$ by constructing a model using the invariant weighted bl...
Let $p \geq 5$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E/K$ be an elliptic curve with good reduction. Let $F_n$ denote the $n$-division polynomial of $E$. Silverman proved that if the reduction is ordinary, then for every $P \in E(K) \setminus \hat{E}(K)$ and a suitable power $q$ of $p$, t...
Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X ; \mathbb{Z}) = 0$. Let $p$ be a prime and suppose that $G = \mathbb{Z}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X ; \mathbb{Z})$). Using equivariant Yang--Mills theory, Hambleton--Lee and...
Let $X$ be the good reduction of a projective hyperk\"ahler variety of dimension $2n\geq4$. We prove that $X$ is quasi-$F$-split if and only if it is Frobenius split, equivalently if $\operatorname{H}^2_{\operatorname{crys}}(X/W)[1/p]$ has a slope-zero part. Thus its quasi-$F$-split height is $1$ or $\infty$. The proof...
Hai-Tao Zou· 0 citations
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